UTM Ideals Varieties and Algorithm - Chapter 1 Section 5 Exercise 11

Problem:

Solution:

Part (a) is really just another way of stating the fundamental theorem of algebra. Note that the set $V(f)$ is simply the set of root of $f$.

Part (b), GCD is 1 $\Leftrightarrow$ these polynomial do not have a common monic factor $\Leftrightarrow$ these polynomial does not have a common root $\Leftrightarrow$ the variety is the empty set.

Part (c), Compute the GCD, and return the variety is empty if the GCD is 1.